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Generalizing Semi-$n$-Potent Rings

Arash Javan, Ahmad Moussavi, Peter Danchev

TL;DR

This work introduces strongly $\\Delta$-tripotent (SDT) rings, where every element is the sum of a tripotent and a $\\Delta(R)$-element that commute, and develops the structural theory of these rings via $\\Delta(R)$ and the Jacobson radical. The central result shows that, modulo $J(R)$, an SDT ring decomposes as the direct product of a Boolean ring and a Yaqub ring, with $R/J(R)$'s components tightly constrained (one reduced and uniquely clean, the other tripotent with $3=0$). The paper also characterizes when triangular matrix rings $T_n(R)$ inherit the SDT property for local rings, reducing to bleached quotient conditions and a surjectivity criterion on certain endomorphisms. Together, these findings extend prior work and provide a framework for generalized SDT phenomena, including potential generalizations to $\\Delta$-n-potent and center-plus-$\\Delta(R)$ variants.

Abstract

We define and explore the class of rings $R$ for which each element in $R$ is a sum of a tripotent element from $R$ and an element from the subring $Δ(R)$ of $R$ which commute each other. Succeeding to obtain a complete description of these rings modulo their Jacobson radical as the direct product of a Boolean ring and a Yaqub ring, our results somewhat generalize those established by Koşan-Yildirim-Zhou in Can. Math. Bull. (2019).

Generalizing Semi-$n$-Potent Rings

TL;DR

This work introduces strongly -tripotent (SDT) rings, where every element is the sum of a tripotent and a -element that commute, and develops the structural theory of these rings via and the Jacobson radical. The central result shows that, modulo , an SDT ring decomposes as the direct product of a Boolean ring and a Yaqub ring, with 's components tightly constrained (one reduced and uniquely clean, the other tripotent with ). The paper also characterizes when triangular matrix rings inherit the SDT property for local rings, reducing to bleached quotient conditions and a surjectivity criterion on certain endomorphisms. Together, these findings extend prior work and provide a framework for generalized SDT phenomena, including potential generalizations to -n-potent and center-plus- variants.

Abstract

We define and explore the class of rings for which each element in is a sum of a tripotent element from and an element from the subring of which commute each other. Succeeding to obtain a complete description of these rings modulo their Jacobson radical as the direct product of a Boolean ring and a Yaqub ring, our results somewhat generalize those established by Koşan-Yildirim-Zhou in Can. Math. Bull. (2019).
Paper Structure (5 sections, 27 theorems, 46 equations)

This paper contains 5 sections, 27 theorems, 46 equations.

Table of Contents

  1. Introduction and Motivation
  2. Examples and Basic Properties
  3. The Main Characterization
  4. Triangular Matrix Rings
  5. Concluding Discussion and Questions

Key Result

Lemma 2.1

(1) Suppose $R = \prod_{i \in I} R_i$. Then, $R$ is an SDT ring if, and only if, for each $i \in I$, $R_i$ is an SDT ring. (2) Suppose $R$ is a ring and $I$ is an ideal of $R$ such that $I \subseteq J(R)$. Then, $R/I$ is an SDT ring.

Theorems & Definitions (56)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • proof
  • Proposition 2.6
  • ...and 46 more